Quaternion Weyl Transform and some uniqueness results

Abstract

In this article, we study the boundedness and several properties of the quaternion Wigner transform. Using the quaternion Wigner transform as a tool, we define the quaternion Weyl transform (QWT) and prove that the QWT is compact for a certain class of symbols in $L^r\left({\mathbb{R}}^4,\mathbb{Q}\right)$ with $1\le r\le 2.$ Moreover, it can not be extended as a bounded operator for symbols in $L^r\left({\mathbb{R}}^4,\mathbb{Q}\right)$ for $2<r<\infty .$ In addition, we prove a rank analogue of the Benedicks-Amrein-Berthier theorem for the QWT. Further, we remark about the set of injectivity and Helgason's support theorem for the quaternion twisted spherical means.

arXiv comment22 pages

References in corpus (4)

• Uniqueness of the group Fourier transform on certain nilpotent Lie groups
• Benedicks-Amrein-Berthier theorem for the Heisenberg motion group and quaternion Heisenberg group
• Unbounded Weyl transform on the Euclidean motion group and Heisenberg motion group
• Spherical means on Métivier groups and support theorem